how to find the square root in simplest form, e.g square root of 18

To find the square root of a number in its simplest form, you break down the number into its prime factors and look for pairs of identical factors, because a pair of identical factors will come out of the square root.

Let's go through the process for \(\sqrt{18}\):

1. **Prime Factorization of 18**:  
   Start by finding the prime factors of 18:  
   \(18 = 2 \times 3 \times 3\).

2. **Look for Pairs of Identical Factors**:  
   In this case, we have a pair of 3's (since \(3 \times 3 = 9\)).

3. **Simplify**:  
   The square root of 18 can be simplified as:
   \[
   \sqrt{18} = \sqrt{3 \times 3 \times 2} = \sqrt{3^2 \times 2}.
   \]
   Now, the square root of \(3^2\) is 3, and we are left with:
   \[
   \sqrt{18} = 3\sqrt{2}.
   \]

So, the simplest form of \(\sqrt{18}\) is \(3\sqrt{2}\).

Would you like more examples, or is there anything else you'd like me to clarify about this process?

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Here are some related questions to explore further:
1. How do you simplify square roots involving larger numbers (like \(\sqrt{72}\))?
2. What if the number is not a perfect square, like \(\sqrt{20}\)?
3. How do you approximate square roots for non-perfect squares?
4. Can you simplify square roots of fractions?
5. What is the process for simplifying cube roots?

Tip: Always start by prime factorizing the number and look for pairs, as this will help simplify the square root quickly!

Learn how to simplify the square root of 18 by prime factorization. This guide demonstrates the process, breaking down 18 into its prime factors and showing how pairs of identical factors simplify the square root. Ideal for students in Grades 7-9.

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